# Thread: Projection of line on plane

1. ## Projection of line on plane

Given $\pi: \begin{cases}l_1: \frac{x-2}{4}\ = \frac{y-1}{2}\ = \frac{z+5}{-4} &\\l_2: \frac{x+4}{-2}\ = \frac{y+1}{0}\ = \frac{z}{1} & \end{cases}$ and the point $M=(1,2,3)$ outside the plane. Find the projection of the line $M_1P$ on plane $\pi$ where $P$ is the intersection point of lines $l_1, l_2$.

All I can do is that I can find the equation of plane $\pi$ but don't have any idea what to do next.

Thank you.

2. Originally Posted by patzer
Given $\pi: \begin{cases}l_1: \frac{x-2}{4}\ = \frac{y-1}{2}\ = \frac{z+5}{-4} &\\l_2: \frac{x+4}{-2}\ = \frac{y+1}{0}\ = \frac{z}{1} & \end{cases}$ and the point $M=(1,2,3)$ outside the plane. Find the projection of the line $M_1P$ on plane $\pi$ where $P$ is the intersection point of lines $l_1, l_2$.
First a comment: No competent textbook or instructor would write a like this one did. It should be $l_2: \frac{x+4}{-2}\ = \frac{z}{1};~y=-1.$.

Now I will not do the work for you.
First show the line intersect in $P$.
The directions of the lines are $D_1:<4,2,-4>~\&~D_2:<-2,0,1>$.
The plane containing $P$ and with normal $N=D_1\times D_2$ is $\pi$.
The projection is the intersection of the plane containing $P$ with normal $\overrightarrow {PM} \times N$ with plane $\pi.$